One modulo N gracefulness of splitting graphs and subdivision of double triangle graphs

V. Ramachandran, C. Sekar

Abstract

A function f is called a graceful labelling of a graph G with q edges if f is an injection fromthe vertices of G to the set {0, 1, 2, . . ., q} such that, when each edge xy is assigned the label|f(x) − f(y)|, the resulting edge labels are distinct. A graph G is said to be one modulo Ngraceful (where N is a positive integer) if there is a function φ from the vertex set of G to{0, 1, N, (N + 1), 2N, (2N + 1), . . . , N(q − 1), N(q − 1) + 1} in such a way that (i) φ is 1 − 1(ii)φ induces a bijection φ* from the edge set of G to {1, N + 1, 2N + 1, . . . , N(q − 1) + 1}where φ*(uv)=|φ(u) − φ(v)| . In this paper we prove that S’(P2n) , S’(P2n+1) , S’(K1,n) , allsubdivision of double triangular snakes (2Δk-snake) and all subdivision of 2mΔk-snake are onemodulo N graceful for all positive integers N.

Keywords

Graceful, One modulo N graceful, Double triangular snakes

References

E.M.Badr and M.E.Abdel-aal, Odd gracefull labling for the subdivision of double triangles graphs,International Journal of Soft Computing, Mathematics and Control (IJSCMC), Vol.2, (2013).